First slide

Functions (XII)

Question

The domain of the function $f (x) = \log_{1 / 2} (x - \frac{1}{2}) + \log_{2} \sqrt{4 x^{2} - 4 x + 5} is$

Moderate

A

$[\frac{1}{2}, \infty)$
B

$(\frac{1}{2}, \infty)$
C

$(- \infty, \infty)$
D

None of these

Solution

$\log_{1 / 2} (x - \frac{1}{2}) is defined if x - \frac{1}{2} > 0 i.e., x > \frac{1}{2} \log_{2} \sqrt{4 x^{2} - 4 x + 5} is defined if 4 x^{2} - 4 x + 5 > 0$
$\Rightarrow 4 [{(x - \frac{1}{2})}^{2} + 1] > 0$ which is true for all real x.
$∴ Domain of f = (\frac{1}{2}, \infty) .$

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